Consider a bond that pays coupon interest of $$45$ at the end of each six$-$month period and that has a face value of $$1000.$ The bond will mature twenty years from today. The bond is currently trading in the market for a price of $$946(94\mathrm{.}6\%$ of face value$).$

$($a$)$ What is the bond$$s yield to maturity? $($Be clear if you are stating the rate as an annual amount or an amount per six months, and about your compounding assumptions$).$

$($b$)$ Suppose now that this bond pays coupon interest in the amount of $$90$ at the end of each year instead of $$45$ at the end of each six$-$month period. The compound annual yield$-$to$-$maturity for the bond is the same as before. That is$,$ we are changing only this one bond characteristic, not the bond$$s default risk or market interest rates. Will the bond with end$-$of$-$year coupon payments have a value that is higher than, lower than, or the same as the bond with semiannual coupon payments? Explain the reasoning for your answer $($no numbers are required to answer this question.

$($c$)$ Compute the specific value of the bond with end$-$of$-$year coupon payments.